Heat the oven and line the dish: Preheat the oven to 425°F. Roll one pie crust into a 9-inch deep-dish pie plate and press it into the corners, leaving the overhang alone for now. Set it in the fridge while you make the filling.

Soften the vegetables: Melt the butter in a large saucepan over medium heat. Add the onion, carrots, and celery and cook for 6-8 minutes, stirring now and then, until the onion is translucent and the carrots have started to soften.

Cook the roux: Stir in the garlic and cook for 30 seconds, then sprinkle in the flour. Stir constantly for a full minute so the flour loses its raw, pasty taste.

Build the gravy: Slowly whisk in the chicken broth, then the half-and-half. Add the thyme, salt, and pepper. Simmer for 4-5 minutes, whisking often, until the gravy is thick enough to coat the back of a spoon.

Add the chicken and peas: Take the pan off the heat and stir in the cooked chicken and the frozen peas. If you poached your own chicken, it should have reached 165°F on a thermometer before you shredded it. Let the filling cool for about 10 minutes.

Assemble the pie: Pour the filling into the chilled bottom crust. Lay the second crust over the top, trim the overhang to about half an inch, fold the edges under, and crimp them together. Cut four or five slits in the center for steam and brush the whole top with the beaten egg.

Bake: Bake for 35-40 minutes, until the crust is deep golden brown and you can see the gravy bubbling through the vents. If the edges brown faster than the center, cover just the rim with a ring of foil.

Rest before slicing: Let the pie sit for 15 minutes before you cut it. The gravy firms up as it cools and the slices come out clean instead of collapsing.